#494304

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1862
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,15,9,12,8,30)(3,16,20,35,21,47,31,45)(4,22,25,40,26,58,32,59)(5,14,28,43,29,44,6,13)(10,27,36,55,34,19,37,18)(11,38,41,60,39,23,42,24)(17,48,51,33,53,63,52,61)(46,56,64,54,62,49,57,50);(1,3,7,20,9,21,8,31)(2,10,15,36,12,34,30,37)(4,17,25,51,26,53,32,52)(5,27,28,55,29,19,6,18)(11,33,41,63,39,61,42,48)(13,45,14,16,43,35,44,47)(22,57,40,46,58,64,59,62)(23,54,24,49,38,50,60,56);(1,4)(2,11)(3,17)(5,24)(6,23)(7,25)(8,32)(9,26)(10,33)(12,39)(13,22)(14,40)(15,41)(16,46)(18,54)(19,56)(20,51)(21,53)(27,49)(28,38)(29,60)(30,42)(31,52)(34,61)(35,64)(36,63)(37,48)(43,58)(44,59)(45,57)(47,62)(50,55);(2,12)(10,34)(11,39)(13,43)(14,44)(15,30)(16,47)(22,58)(33,61)(35,45)(36,37)(40,59)(41,42)(46,62)(48,63)(57,64)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401500);